Solve frac{w^-frac{12{13}}}{w^frac{12{13}}*w^-frac{4{13}}}

$$\frac { w ^ { - \frac { 12 } { 13 } } } { w ^ { \frac { 12 } { 13 } } \cdot w ^ { - \frac { 4 } { 13 } } }$$

Differentiate w.r.t. w

$$-\frac{20}{13w^{\frac{33}{13}}}$$

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add \frac{12}{13} and -\frac{4}{13} to get \frac{8}{13}.

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{w^{-\frac{12}{13}}}{w^{\frac{8}{13}}})$$

Rewrite w^{\frac{8}{13}} as w^{-\frac{12}{13}}w^{\frac{20}{13}}. Cancel out w^{-\frac{12}{13}} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{1}{w^{\frac{20}{13}}})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(w^{\frac{20}{13}}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}w}(w^{\frac{20}{13}})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-\left(w^{\frac{20}{13}}\right)^{-2}\times \frac{20}{13}w^{\frac{20}{13}-1}$$

Simplify.

$$-\frac{20}{13}w^{\frac{7}{13}}\left(w^{\frac{20}{13}}\right)^{-2}$$

Evaluate

$$\frac{1}{w^{\frac{20}{13}}}$$